Introduction – In the last decades the non-equilibrium thermodynamics has received a great attention [1]}, [2]}, [3]}. Among the several results achieved in this field, the fluctuation theorems (see, e.g., Ref. [4]} for a review) undoubtedly are of fundamental importance. For instance, these theorems lead to the Jarzynski equality [5]} and the second law of thermodynamics. Recently, it is shown how, for a cyclic and time-reversal symmetric protocol, these theorems imply the so-called thermodynamic uncertainty relations [6]}, [7]}, [8]}, i.e. they give a lower bound of fluctuations of observables, which involves the entropy produced in the process. Conversely, for the case of general fluctuation theorems, hysteretic thermodynamic uncertainty relations [9]}, [10]} and a lower bound for the entropy produced in the forward process [11]} are derived.
